Theorem cevathlem1 41056

Description: Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Hypotheses

Ref	Expression
cevathlem1.a	|- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
cevathlem1.b	|- ( ph -> ( D e. CC /\ E e. CC /\ F e. CC ) )
cevathlem1.c	|- ( ph -> ( G e. CC /\ H e. CC /\ K e. CC ) )
cevathlem1.d	|- ( ph -> ( A =/= 0 /\ E =/= 0 /\ C =/= 0 ) )
cevathlem1.e	|- ( ph -> ( ( A x. B ) = ( C x. D ) /\ ( E x. F ) = ( A x. G ) /\ ( C x. H ) = ( E x. K ) ) )

Assertion

Ref	Expression
cevathlem1	|- ( ph -> ( ( B x. F ) x. H ) = ( ( D x. G ) x. K ) )

Proof of Theorem cevathlem1

Step	Hyp	Ref	Expression
1		cevathlem1.a	. . . . 5 |- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
2	1	simp2d 1074	. . . 4 |- ( ph -> B e. CC )
3		cevathlem1.b	. . . . 5 |- ( ph -> ( D e. CC /\ E e. CC /\ F e. CC ) )
4	3	simp3d 1075	. . . 4 |- ( ph -> F e. CC )
5	2, 4	mulcld 10060	. . 3 |- ( ph -> ( B x. F ) e. CC )
6		cevathlem1.c	. . . 4 |- ( ph -> ( G e. CC /\ H e. CC /\ K e. CC ) )
7	6	simp2d 1074	. . 3 |- ( ph -> H e. CC )
8	5, 7	mulcld 10060	. 2 |- ( ph -> ( ( B x. F ) x. H ) e. CC )
9	3	simp1d 1073	. . . 4 |- ( ph -> D e. CC )
10	6	simp1d 1073	. . . 4 |- ( ph -> G e. CC )
11	9, 10	mulcld 10060	. . 3 |- ( ph -> ( D x. G ) e. CC )
12	6	simp3d 1075	. . 3 |- ( ph -> K e. CC )
13	11, 12	mulcld 10060	. 2 |- ( ph -> ( ( D x. G ) x. K ) e. CC )
14	1	simp1d 1073	. . . 4 |- ( ph -> A e. CC )
15	3	simp2d 1074	. . . 4 |- ( ph -> E e. CC )
16	14, 15	mulcld 10060	. . 3 |- ( ph -> ( A x. E ) e. CC )
17	1	simp3d 1075	. . 3 |- ( ph -> C e. CC )
18	16, 17	mulcld 10060	. 2 |- ( ph -> ( ( A x. E ) x. C ) e. CC )
19		cevathlem1.d	. . . . 5 |- ( ph -> ( A =/= 0 /\ E =/= 0 /\ C =/= 0 ) )
20	19	simp1d 1073	. . . 4 |- ( ph -> A =/= 0 )
21	19	simp2d 1074	. . . 4 |- ( ph -> E =/= 0 )
22	14, 15, 20, 21	mulne0d 10679	. . 3 |- ( ph -> ( A x. E ) =/= 0 )
23	19	simp3d 1075	. . 3 |- ( ph -> C =/= 0 )
24	16, 17, 22, 23	mulne0d 10679	. 2 |- ( ph -> ( ( A x. E ) x. C ) =/= 0 )
25		cevathlem1.e	. . . . . . . 8 |- ( ph -> ( ( A x. B ) = ( C x. D ) /\ ( E x. F ) = ( A x. G ) /\ ( C x. H ) = ( E x. K ) ) )
26	25	simp1d 1073	. . . . . . 7 |- ( ph -> ( A x. B ) = ( C x. D ) )
27	25	simp2d 1074	. . . . . . 7 |- ( ph -> ( E x. F ) = ( A x. G ) )
28	26, 27	oveq12d 6668	. . . . . 6 |- ( ph -> ( ( A x. B ) x. ( E x. F ) ) = ( ( C x. D ) x. ( A x. G ) ) )
29	14, 2, 15, 4	mul4d 10248	. . . . . 6 |- ( ph -> ( ( A x. B ) x. ( E x. F ) ) = ( ( A x. E ) x. ( B x. F ) ) )
30	17, 9, 14, 10	mul4d 10248	. . . . . 6 |- ( ph -> ( ( C x. D ) x. ( A x. G ) ) = ( ( C x. A ) x. ( D x. G ) ) )
31	28, 29, 30	3eqtr3d 2664	. . . . 5 |- ( ph -> ( ( A x. E ) x. ( B x. F ) ) = ( ( C x. A ) x. ( D x. G ) ) )
32	25	simp3d 1075	. . . . 5 |- ( ph -> ( C x. H ) = ( E x. K ) )
33	31, 32	oveq12d 6668	. . . 4 |- ( ph -> ( ( ( A x. E ) x. ( B x. F ) ) x. ( C x. H ) ) = ( ( ( C x. A ) x. ( D x. G ) ) x. ( E x. K ) ) )
34	16, 5, 17, 7	mul4d 10248	. . . 4 |- ( ph -> ( ( ( A x. E ) x. ( B x. F ) ) x. ( C x. H ) ) = ( ( ( A x. E ) x. C ) x. ( ( B x. F ) x. H ) ) )
35	17, 14	mulcld 10060	. . . . 5 |- ( ph -> ( C x. A ) e. CC )
36	35, 11, 15, 12	mul4d 10248	. . . 4 |- ( ph -> ( ( ( C x. A ) x. ( D x. G ) ) x. ( E x. K ) ) = ( ( ( C x. A ) x. E ) x. ( ( D x. G ) x. K ) ) )
37	33, 34, 36	3eqtr3d 2664	. . 3 |- ( ph -> ( ( ( A x. E ) x. C ) x. ( ( B x. F ) x. H ) ) = ( ( ( C x. A ) x. E ) x. ( ( D x. G ) x. K ) ) )
38	14, 15, 17	mul32d 10246	. . . . 5 |- ( ph -> ( ( A x. E ) x. C ) = ( ( A x. C ) x. E ) )
39	14, 17	mulcomd 10061	. . . . . 6 |- ( ph -> ( A x. C ) = ( C x. A ) )
40	39	oveq1d 6665	. . . . 5 |- ( ph -> ( ( A x. C ) x. E ) = ( ( C x. A ) x. E ) )
41	38, 40	eqtrd 2656	. . . 4 |- ( ph -> ( ( A x. E ) x. C ) = ( ( C x. A ) x. E ) )
42	41	oveq1d 6665	. . 3 |- ( ph -> ( ( ( A x. E ) x. C ) x. ( ( D x. G ) x. K ) ) = ( ( ( C x. A ) x. E ) x. ( ( D x. G ) x. K ) ) )
43	37, 42	eqtr4d 2659	. 2 |- ( ph -> ( ( ( A x. E ) x. C ) x. ( ( B x. F ) x. H ) ) = ( ( ( A x. E ) x. C ) x. ( ( D x. G ) x. K ) ) )
44	8, 13, 18, 24, 43	mulcanad 10662	1 |- ( ph -> ( ( B x. F ) x. H ) = ( ( D x. G ) x. K ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  cevath  41058